Computer Science > Data Structures and Algorithms

Title:Streaming Algorithms from Precision Sampling

Abstract: A technique introduced by Indyk and Woodruff [STOC 2005] has inspired several
recent advances in data-stream algorithms. We show that a number of these
results follow easily from the application of a single probabilistic method
called Precision Sampling. Using this method, we obtain simple data-stream
algorithms that maintain a randomized sketch of an input vector
$x=(x_1,...x_n)$, which is useful for the following applications. 1) Estimating
the $F_k$-moment of $x$, for $k>2$. 2) Estimating the $\ell_p$-norm of $x$, for
$p\in[1,2]$, with small update time. 3) Estimating cascaded norms
$\ell_p(\ell_q)$ for all $p,q>0$. 4) $\ell_1$ sampling, where the goal is to
produce an element $i$ with probability (approximately) $|x_i|/\|x\|_1$. It
extends to similarly defined $\ell_p$-sampling, for $p\in [1,2]$.
For all these applications the algorithm is essentially the same: scale the
vector x entry-wise by a well-chosen random vector, and run a heavy-hitter
estimation algorithm on the resulting vector. Our sketch is a linear function
of x, thereby allowing general updates to the vector x.
Precision Sampling itself addresses the problem of estimating a sum
$\sum_{i=1}^n a_i$ from weak estimates of each real $a_i\in[0,1]$. More
precisely, the estimator first chooses a desired precision $u_i\in(0,1]$ for
each $i\in[n]$, and then it receives an estimate of every $a_i$ within additive
$u_i$. Its goal is to provide a good approximation to $\sum a_i$ while keeping
a tab on the "approximation cost" $\sum_i (1/u_i)$. Here we refine previous
work [Andoni, Krauthgamer, and Onak, FOCS 2010] which shows that as long as
$\sum a_i=\Omega(1)$, a good multiplicative approximation can be achieved using
total precision of only $O(n\log n)$.